Entanglement tongue and quantum synchronization of
disordered oscillators
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Lee, Tony E., Ching-Kit Chan, and Shenshen Wang.
“Entanglement Tongue and Quantum Synchronization of
Disordered Oscillators.” Phys. Rev. E 89, no. 2 (February 2014).
© 2014 American Physical Society
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http://dx.doi.org/10.1103/PhysRevE.89.022913
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Detailed Terms
PHYSICAL REVIEW E 89, 022913 (2014)
Entanglement tongue and quantum synchronization of disordered oscillators
Tony E. Lee,1,2 Ching-Kit Chan,1,2 and Shenshen Wang3
1
ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA
2
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
3
Department of Physics and Department of Chemical Engineering, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA
(Received 16 December 2013; published 12 February 2014)
We study the synchronization of dissipatively coupled van der Pol oscillators in the quantum limit, when
each oscillator is near its quantum ground state. Two quantum oscillators with different frequencies exhibit
an entanglement tongue, which is the quantum analog of an Arnold tongue. It means that the oscillators are
entangled in steady state when the coupling strength is greater than a critical value, and the critical coupling
increases with detuning. An ensemble of many oscillators with random frequencies still exhibits a synchronization
phase transition in the quantum limit, and we analytically calculate how the critical coupling depends on the
frequency disorder. Our results can be experimentally observed with trapped ions or neutral atoms.
DOI: 10.1103/PhysRevE.89.022913
PACS number(s): 05.45.Xt, 42.50.Lc, 03.65.Ud
I. INTRODUCTION
(a)
(b)
0.16
1539-3755/2014/89(2)/022913(10)
5
50
0.12
40
V (units of κ1)
4
V (units of κ1)
Synchronization is a fascinating phenomenon at the interface of statistical physics and nonlinear dynamics [1,2].
It is a collective behavior that arises among a group of
self-sustained oscillators, each with a random intrinsic frequency. The interaction between the oscillators overcomes the
frequency disorder and causes them to oscillate in unison.
Synchronization of biological cells plays an important role in
heart beats [3], circadian rhythm [4], and neural networks [5].
It is also important in active hydrodynamic systems [6],
such as the beating of flagella [7] and arrays of cilia [8].
Applications of synchronization include the self-organization
of laser arrays [9], improving the frequency precision of
oscillators [10], and stabilizing atomic clocks with each
other [11].
There has been much theoretical work on synchronization
of classical oscillators. Each oscillator is usually modeled as
a nonlinear dynamical system with a limit-cycle solution that
oscillates with its own intrinsic frequency. Then due to the
mutual interaction, the oscillators spontaneously synchronize
with each other in steady state.
Synchronization is usually studied in two scenarios: two
oscillators and a large ensemble of oscillators with all-to-all
coupling. In the case of two oscillators, phase locking occurs
when the coupling strength is above a critical value [1,12].
This critical coupling increases with the oscillators’ frequency
detuning. The “Arnold tongue” refers to the set of coupling and
detuning values for which phase locking occurs [Fig. 1(a)].
In the case of a large ensemble of oscillators with random
frequencies, there is a nonequilibrium phase transition from
the unsynchronized phase to the synchronized phase [13–19].
The critical coupling for the phase transition depends on the
frequency disorder.
There has been growing interest in synchronization of
quantum systems [20–26]. In this case, each oscillator is a
quantum harmonic oscillator with driving, dissipation, and
nonlinearity. The classical limit corresponds to when each
oscillator has many phonons (or photons), while the quantum
limit corresponds to when each oscillator is near its quantum
ground state. Quantum mechanics introduces two effects. The
first is quantum noise, which is due to the oscillator gaining or
locked
3
entangled
30
2
0.08
20
0.04
1
0
unlocked
−10
−5
0
5
Δ (units of κ )
1
10
10
0
unentangled
−100
−50
0
50
100
0
Δ (units of κ )
1
FIG. 1. (Color online) (a) Arnold tongue for phase locking of
two classical oscillators. V is the coupling strength, and is the
difference of the intrinsic frequencies. (b) Entanglement tongue for
two oscillators in the quantum limit. Concurrence is plotted using
color scale on right. The dashed line marks the edge of the entangled
region.
losing individual phonons [27]. The second effect is that the
oscillators can be quantum mechanically entangled with each
other [28]. The general question is whether synchronization
survives in the quantum limit, and how quantum mechanics
qualitatively changes the behavior.
To study quantum synchronization, it is useful to consider
quantum van der Pol oscillators [22,23], since there has been
a lot of work on the synchronization of classical van der Pol
oscillators [1,12,16–19]. Recently, one of us showed that when
the quantum oscillators are reactively coupled (via a term in
the Hamiltonian), there is no synchronization or entanglement
in the quantum limit, when the oscillators are confined to the
ground state |0 and the single-phonon state |1 [22]. This is
because quantum noise washes out the phase correlations.
In this paper, we study the synchronization of dissipatively
coupled van der Pol oscillators in the quantum limit, and we
find significant differences with the reactive case. We first
consider the case of two quantum oscillators. We find that
the oscillators still exhibit phase correlations in the quantum
limit, when each oscillator occupies only |0 and |1. Also, the
oscillators exhibit entanglement, which is a genuinely quantum
022913-1
©2014 American Physical Society
TONY E. LEE, CHING-KIT CHAN, AND SHENSHEN WANG
PHYSICAL REVIEW E 89, 022913 (2014)
property and can be quantified using concurrence [29]. In order
for entanglement to exist in steady state, the coupling strength
must be larger than a critical value, and the critical coupling
increases with the detuning of the oscillators [Fig. 1(b)]. This
“entanglement tongue” is the quantum analog of the Arnold
tongue.
Then we consider a large ensemble of quantum oscillators
with random frequencies and all-to-all coupling. The synchronization phase transition still occurs in the quantum limit, but
each oscillator must occupy at least the two-phonon state |2.
We analytically calculate the critical coupling by doing a linear
stability analysis of the unsynchronized phase and find good
agreement with numerical simulations. The dependence of the
critical coupling on the frequency disorder is different in the
quantum limit than in the classical limit.
In Ref. [26], it was found that linear oscillators can exhibit
collective oscillations and entanglement when all normal
modes except one are damped. In contrast, we consider nonlinear oscillators in order to directly compare with well-known
classical synchronization models. Also, Ref. [23] considered
the synchronization of one quantum van der Pol oscillator with
an external drive and found that quantum noise introduces a
nonzero threshold of driving strength, below which there is
only weak frequency locking. In contrast, we consider two or
more oscillators and focus on phase locking.
In Sec. II, we describe the model of quantum van der
Pol oscillators, as well as experimental implementation with
cold atoms. In Sec. III, we consider two quantum oscillators
and characterize their phase correlations and entanglement.
In Sec. IV, we consider a large ensemble of quantum
oscillators and characterize the synchronization transition. In
Sec. V, we conclude and suggest future directions of research.
The Appendix provides details on the two-mode Wigner
function.
II. MODEL
A. Classical regime
A common starting point for studying classical synchronization is to consider an oscillator whose complex amplitude
α obeys
α̇ = −iωα + α(κ1 − 2κ2 |α|2 ).
(1)
This is the amplitude equation for the van der Pol oscillator [1,30]. It is also the normal form for a Hopf bifurcation [31].
κ1 and κ2 correspond to negative damping and nonlinear
damping, respectively. They are assumed to be positive, so
the
κ1
steady-state solution is a limit cycle with amplitude |α| = 2κ
2
and frequency ω. The phase of the oscillations is free, since
Eq. (1) is time-translationally invariant.
Now consider two coupled oscillators with linear dissipative coupling:
α̇1 = −iω1 α1 + α1 (κ1 − 2κ2 |α1 |2 ) + V (α2 − α1 ),
(2)
α̇2 = −iω2 α2 + α2 (κ1 − 2κ2 |α2 |2 ) + V (α1 − α2 ).
(3)
This model is well-known in nonlinear dynamics [1,12]. It is
convenient to use polar coordinates: αn = rn eiθn . Then Eqs. (2)
and (3) become
r˙1 = r1 κ1 − 2κ2 r12 − V (r1 − r2 cos θ ),
r˙2 = r2 κ1 − 2κ2 r22 − V (r2 − r1 cos θ ),
r2
r1
sin θ,
+
θ̇ = − − V
r2
r1
(4)
(5)
(6)
where θ = θ2 − θ1 is the phase difference, and = ω2 − ω1
is the detuning. Equation (6) shows that the coupling causes
the oscillator phases to attract each other, while the detuning
pulls them apart. If V is larger than a critical value Vc , the
oscillators phase lock with each other, i.e., there is a stable
fixed-point solution with r1 , r2 , θ constant in time. If V <
Vc , the oscillators are unlocked and |θ | increases over time.
Vc increases with ||, which reflects the fact that the more
different the oscillators are, the harder it is to synchronize
them. The “Arnold tongue” is the region in V , space such
that V > Vc (), as shown in Fig. 1(a) [1,12].
Then consider the generalization to N oscillators with allto-all coupling:
α̇n = −iωn αn + αn (κ1 − 2κ2 |αn |2 ) +
N
V (αm − αn ), (7)
N m=1
where the frequencies ωn are randomly drawn from a distribution g(ω), and we let N → ∞. This model has also been
studied previously [16,17,32,33]. The competition between the
interaction and the frequency disorder results in a continuous
phase transition when the coupling is at a critical value
Vc (which is not the same as for two oscillators). When
V > Vc , the interaction overcomes the frequency disorder,
and there
is macroscopic synchronization. The order parameter
|(1/N ) n αn | is zero in the unsynchronized phase and greater
than zero in the synchronized phase. The synchronized phase
breaks the U (1) symmetry present in the system (αn → αn eiβ ).
Of interest is how Vc depends on the frequency disorder g(ω).
In the limit of small V , Eq. (7) is equivalent to the well-known
Kuramoto model [13–15].
B. Quantum regime
We are interested in what happens to the synchronization
behavior of the above models in the quantum limit, when
the oscillators are near the ground state. To study this, we
quantize the above classical models. First, we review how
to quantize the van der Pol oscillator in Eq. (1). We let the
oscillator be a quantum harmonic oscillator, meaning that it
exists in the Hilbert space of Fock states {|n}, where n is
the number of phonons. The quantum van der Pol oscillator is
described in terms of a master equation for the density matrix
ρ [22,23]:
ρ̇ = −i[H,ρ] + κ1 (2a † ρa − aa † ρ − ρaa † )
+ κ2 (2a 2 ρa †2 − a †2 a 2 ρ − ρa †2 a 2 ),
†
H = ωa a,
(8)
(9)
where = 1. There are two dissipative processes: the oscillator gains one phonon at a time with rate 2κ1 aa † , and it loses
two phonons at a time with rate 2κ2 a †2 a 2 . These correspond
to negative damping and nonlinear damping, respectively.
022913-2
ENTANGLEMENT TONGUE AND QUANTUM . . .
PHYSICAL REVIEW E 89, 022913 (2014)
This model has also been studied in the context of polariton
condensates [34,35]. Other dissipative models were similarly
quantized in Refs. [36–40].
The quantum limit corresponds to when κ2 is large relative
to κ1 , since then the oscillator is near the ground state [22].
Conversely, the classical limit corresponds to when κ2 → 0,
since then the oscillator has an infinite number of phonons. To
see the connection with the classical model, one notes from
Eq. (8) that
da
(10)
= −iωa + κ1 a − 2κ2 a † a 2 .
dt
In the classical limit, one can replace the operator a with a
complex number α denoting a coherent state, and Eq. (10)
becomes Eq. (1).
An important feature of the quantum model is that the limitcycle solution survives in the quantum limit. This can be seen
by solving for the steady state of Eq. (8) in the limit κ2 →
∞ [22]:
ρ=
2
|00|
3
+
1
|11|.
3
(11)
The oscillator is confined to |0 and |1, since any higher Fock
state is immediately annihilated by the nonlinear damping.
[For large κ2 , the population in |n is ∼ O(1/κ2n−1 ) for n 2.]
In this limit, a † a is still nonzero, since the population is not
entirely in the ground state. Also, the phase of the oscillator is
free, since ρ has no off-diagonal elements. Thus, Eq. (11) can
still be considered a limit cycle. Another way to see this is to
plot the Wigner function corresponding to Eq. (11); the Wigner
function has a ring shape, just like it would for a classical limit
cycle [22]. (Note that although the oscillator is effectively a
two-level system in this limit, it is still an oscillator in the
sense that it has a Wigner function. Also note that the Wigner
function is positive.)
The survival of the limit cycle is due to the nonlinear
damping in Eq. (8). If, on the other hand, the damping were
linear, the limit cycle would not survive in the quantum limit,
since the oscillator would be entirely in the ground state [41].
But since the the limit cycle does survive, it makes sense to
talk about synchronization of multiple oscillators in this limit.
Synchronization due to reactive coupling (via a term in the
Hamiltonian) was discussed in Ref. [22]. Here, we consider
dissipative coupling.
Consider two quantum oscillators, a1 and a2 . The quantum
version of Eqs. (2) and (3) is
(2an† ρan − an an† ρ − ρan an† )
ρ̇ = −i[H,ρ] + κ1
n
+ κ2
2an2 ρan†2 − an†2 an2 ρ − ρan†2 an2
n
H =
+ V (2cρc† − c† cρ − ρc† c),
(12)
†
ω1 a1 a1
(13)
+
†
ω2 a2 a2 ,
where c = a1 − a2 is the jump operator that leads to dissipative
coupling [42–44]. It is easy to show that Eq. (12) reproduces
Eqs. (2) and (3) in the classical limit. To understand the
coupling in the quantum model, it is useful √
to note that the
symmetric superposition |S = (|01 + |10)/ 2 corresponds
to the in-phase state, while the antisymmetric superposition
√
|A = (|01 − |10)/ 2 corresponds to the antiphase state.
(The reason for this correspondence is discussed in the
Appendix.) |S is a dark state with respect to c, since c|S = 0.
Thus, c dissipatively pumps the system into |S, leading to
in-phase locking [42–44]. We analyze this model in Sec. III.
Then consider the quantum version of the all-to-all model
in Eq. (7) for N oscillators:
(2an† ρan − an an† ρ − ρan an† )
ρ̇ = −i[H,ρ] + κ1
+ κ2
n
2an2 ρan†2
− an†2 an2 ρ − ρan†2 an2
n
V †
†
†
+
(2cmn ρcmn
− cmn
cmn ρ − ρcmn
cmn ),
N m<n
H =
ωn an† an ,
(14)
(15)
n
where cmn = am − an , and the frequencies ωn are randomly
drawn from a distribution g(ω). We are interested in the limit
N → ∞. We analyze this model in Sec. IV.
C. Experimental implementation
In the limit κ2 → ∞, Eqs. (12) and (13) can be mapped to
a dissipative spin model, where |0 and |1 correspond to | ↓
and | ↑, respectively:
(2σn+ ρσn− − σn− σn+ ρ − ρσn− σn+ )
ρ̇ = −i[H,ρ] + κ1
n
+ 2κ1
(2σn− ρσn+ − σn+ σn− ρ − ρσn+ σn− )
n
+ V (2c̃ρ c̃† − c̃† c̃ρ − ρ c̃† c̃),
(16)
(17)
H = ω1 σ1+ σ1− + ω2 σ2+ σ2− ,
−
−
where c̃ = σ1 − σ2 . The Pauli operators are defined as
σn− = |01|n and σn+ = |10|n . The reason for this mapping
4κ1
4κ2
is that the transitions |1 −→ |2 −→ |0 can be viewed as
4κ1
|1 −→ |0 in the limit κ2 → ∞. In the spin model, each spin
is effectively coupled to a bath with nonzero temperature,
which incoherently excites and de-excites the spins [27].
A model similar to Eqs. (16) and (17) can be implemented
using two trapped ions. Each spin corresponds to an ion, and
| ↓ and | ↑ correspond to the ground and excited states of
an electronic transition, respectively. One would drive the
electronic transition using incoherent light to mimic a finitetemperature bath. Then one would implement a nonlinear
dissipative coupling c̃ = (σ1+ + σ2+ )(σ1− − σ2− ) by means of
digital quantum simulation as demonstrated in Ref. [44]. Such
a nonlinear coupling also pumps the system into |S, leading
to in-phase locking. This scheme can be generalized to N > 2
with all-to-all coupling. In this paper, we discuss only linear
dissipative coupling in order to connect with previous works
on the classical model, but we have checked that nonlinear
dissipative coupling leads to similar results.
An alternative approach to implementing Eqs. (16) and (17)
is to use atoms within an optical cavity as explained in
Ref. [24]. The cavity mediates a linear dissipative coupling
between the atoms with c̃ = σ1− + σ2− , which pumps the
022913-3
TONY E. LEE, CHING-KIT CHAN, AND SHENSHEN WANG
PHYSICAL REVIEW E 89, 022913 (2014)
system into |A, leading to antiphase locking. In addition,
there are other methods (with Rydberg atoms [45] or trapped
ions [46]) to dissipatively pump the system into |A, although
the resulting c̃ is more complicated.
1
(a)
0.2
(b)
0.8
W
W
0.6
0.15
0.4
III. TWO QUANTUM OSCILLATORS
In this section, we study the quantum model defined by
Eqs. (12) and (13). There are several factors at work here.
As in the classical model, the dissipative coupling causes the
phases to attract and lock with each other, while the detuning
inhibits phase locking. But the quantum model introduces two
new features. The first new feature is quantum noise due to the
stochastic dissipation that adds one phonon (κ1 ) and removes
two phonons (κ2 ) at a time. Quantum noise is quantitatively
equivalent to classical white noise when an oscillator has
many phonons, but not when the oscillator is near the ground
state, since then the addition or loss of a phonon has a large
effect on the system [27]. We are interested in whether phase
attraction survives in the quantum limit, when there is a lot
of quantum noise. The second new feature is entanglement,
which results from the fact that the dissipative coupling tries to
pump the system into the entangled state |S. We are interested
in whether entanglement exists in the quantum limit despite
decoherence from quantum noise and frequency detuning.
A. Phase correlations
We want to find the steady-state density matrix of Eq. (12).
Since we only care about the quantum limit (κ2 → ∞), we
can use a truncated Hilbert space that contains only |0 and
|1. Then the steady-state density matrix in the limit κ2 → ∞
is
κ1 (5κ1 + 2V )[2 + 4(3κ1 + V )2 ]
,
(18)
N
κ1 (2κ1 + V )[2 + 4(3κ1 + V )2 ]
01|ρ|01 = 10|ρ|10 =
,
N
(19)
00|ρ|00 = 1 −
κ12 [2 + 4(3κ1 + V )2 ]
,
(20)
N
2κ1 V (κ1 + V )(6κ1 + 2V − i)
01|ρ|10 = 10|ρ|01∗ =
,
N
(21)
2
2
N = (3κ1 + V ) 3κ1 + 36κ1
(22)
+ 2 + 108κ12 V + 32κ1 V 2 ,
11|ρ|11 =
and the other matrix elements are zero. [This result is most
easily derived using the spin model in Eqs. (16) and (17).]
The fact that there are off-diagonal matrix elements like
|0110| means that there are still phase correlations between
the oscillators in the quantum limit.
The phase correlations can be better understood from
the two-mode Wigner function W (α1 ,α1∗ ,α2 ,α2∗ ), which is a
quasiprobability density for the two-oscillator system [47].
Let us use radial coordinates: αn = rn eiθn . Since we only
care about the relative phase θ = θ2 − θ1 , we integrate out
0.2
0.1
−π
−π/2
0
θ2 − θ1
π/2
π
0
−π
−π/2
0
θ2 − θ1
π/2
π
FIG. 2. (Color online) Two-mode Wigner function in (a) the
quantum limit (κ2 → ∞) and (b) the classical limit (κ2 → 0), plotted
as a function of phase difference with V = 3κ1 : = 0 (solid, black
line) and = 4κ1 (dashed, red line).
r1 , r2 , θ1 + θ2 so that W depends only on θ :
W (θ ) =
1
κ1 V (κ1 + V )[2(3κ1 + V ) cos θ − sin θ ]
+
.
2π
N
(23)
Details of this calculation are provided in the Appendix.
Figure 2(a) shows example Wigner functions in the quantum
limit. The Wigner function is peaked at some value of θ ,
meaning that the relative phase tends toward some value. The
peak is quite broad, indicating that the phase correlation is
imperfect. If the oscillators are identical ( = 0), then the peak
is at θ = 0, which means the oscillators tend to be in-phase
with each other. However, if they are nonidentical ( = 0), the
peak is offset from θ = 0: when > 0, then θ < 0. The height
of the peak increases with V due to stronger phase attraction.
This behavior is actually similar to the classical model
[Eqs. (2) and (3)]. When classical oscillators are phase locked,
the Wigner function has a delta-function peak at a certain value
of θ , as seen in Fig. 2(b). If > 0, then the peak would be at
θ < 0, just like in Eq. (23).1 Then suppose one added white
noise to Eqs. (2) and (3). The noise would broaden the peaks in
the Wigner function, just like in Fig. 2(a). The oscillator would
no longer be strictly phase locked due to noise-induced phase
slips, but there would still be a tendency toward locking [1].
There would also no longer be an Arnold tongue due to the
absence of phase locking.
Thus, the phase correlations in the quantum limit are still
qualitatively consistent with finite classical noise. This is a
rather surprising result, since naı̈vely one would expect quantum noise to completely wash out the phase correlations when
κ2 → ∞. In fact, when the coupling is reactive [via a term
†
†
in the Hamiltonian, V (a1 a2 + a1 a2 )] instead of dissipative,
there are indeed no phase correlations in this limit [22]. So
the behavior in the quantum limit depends a great deal on the
type of coupling: phase attraction survives with dissipative
coupling but not with reactive coupling.
1
Equations (2) and (3) are usually written with the opposite sign
convention for ωn [1,12]. With the usual convention, when the
oscillators are locked, the oscillator with the larger frequency is ahead
in phase. Then using the sign convention of Eqs. (2) and (3), if > 0,
the locked state has θ < 0.
022913-4
ENTANGLEMENT TONGUE AND QUANTUM . . .
PHYSICAL REVIEW E 89, 022913 (2014)
There is another similarity between the quantum and
classical behavior. The classical model can exhibit “amplitude
death” (or “oscillator death”), which means that the trivial
solution α1 = α2 = 0 is stable [1,12]. This occurs when ||
and V are large and is due to the fact that the dissipative
coupling increases the overall dissipation each oscillator
feels. Interestingly, the quantum model exhibits something
akin to amplitude death: in the limit ||,V → ∞, we find
†
†
a1 a1 ,a2 a2 → 0.
B. Entanglement tongue
Now we check whether the quantum model exhibits
entanglement, which is a property unique to quantum systems.
Two particles are entangled if and only if their density matrix
cannot be written as a sum of separable states [28]. Since
each oscillator is limited to |0 and |1 in the quantum
limit, we can quantify the entanglement by calculating the
concurrence for the density matrix ρ in Eqs. (18)–(21).
Consider ρ̃ = (σy ⊗ σy )ρ ∗ (σy ⊗ σy ). Concurrence is defined
as C ≡ max(0,λ1 − λ2 − λ3 − λ4 ), where λ1 ,λ2 ,λ3 ,λ4 are the
square roots of the eigenvalues of ρ ρ̃ in decreasing order [29].
C ∈ [0,1] by definition and the oscillators are entangled if and
only if C > 0. Also, larger C implies more entanglement.
Whether entanglement exists is the result of several competing processes. The dissipative coupling tries to entangle
the oscillators since it pumps the system into |S. However,
the negative damping (κ1 ) and nonlinear damping (κ2 ) lead to
decoherence since they act on individual oscillators. Also, the
detuning leads to dephasing between the oscillators and thus
reduces the entanglement.
The concurrence for Eqs. (18)–(21) is plotted in Fig. 1(b).
There is indeed entanglement, but only when V is greater than
a critical value Vc that depends on . When the oscillators are
identical ( = 0), Vc = 8.664κ1 . This is nonzero because the
coupling must be strong enough to overcome decoherence
from the onsite dissipation (κ1 and κ2 ). As || increases,
Vc increases, because the coupling must overcome additional
decoherence from detuning. This is reminiscent of how the
critical coupling for phase locking in the classical model
increases with ||, giving rise to an Arnold tongue. Thus,
we call the entanglement region defined by V > Vc () the
“entanglement tongue.” It reflects the fact that the more
different the oscillators are, the harder it is to entangle them.
In the limit of large ||, Vc = ||
. Also, the maximum C
2
1
is 4 , which occurs when = 0 and V = ∞.
To summarize the results for two oscillators, the quantum
behavior resembles the classical behavior in terms of phase
correlations, but the quantum model exhibits entanglement,
which is a genuinely quantum feature. Although there is no
critical coupling for phase locking in the quantum limit, there
is a critical coupling for entanglement. As such, the Arnold
tongue is replaced by the entanglement tongue.
disorder g(ω). As in Sec. III, there are several factors at
work. The coupling tries to synchronize the oscillators with
each other, but the frequency disorder and quantum noise
inhibit synchronization. In this section, we do not discuss
bipartite entanglement, since the all-to-all coupling leads
to a concurrence that scales as ∼ 1/N, but it would be
interesting, as a future work, to see whether there is multipartite
entanglement.
The strategy for calculating Vc is similar to that for classical
models [15,17,19]. First we find the mean-field equations.
Then we find the unsynchronized state. Then we do a linear
stability analysis around the unsynchronized state. The onset
of instability of the unsynchronized state signals a continuous
phase transition to the synchronized state. The key difference
with classical models is that the stability analysis here is based
on the master equation for the density matrix instead of a
partial differential equation for the probability density.
With dissipative coupling, the phase transition turns out to
be continuous. In contrast, with reactive coupling, the phase
transition is discontinuous [22].
A. Mean-field equations
We want to rewrite Eq. (14) in a way such that each
oscillator interacts with the mean field. We first make the
mean-field ansatz that the density matrix
is a product state of
density matrices for each site: ρ = n ρn [43]. (This ansatz
is exact when N is infinite, as can be shown rigorously using
a phase-space approach [48].) Then we plug this ansatz into
Eq. (14). Using the fact that
ρ̇ = ρ̇1 ⊗ ρ2 ⊗ ρ3 ⊗ · · · + ρ1 ⊗ ρ̇2 ⊗ ρ3 ⊗ · · ·
+ ρ1 ⊗ ρ2 ⊗ ρ̇3 ⊗ · · · + · · · ,
(24)
tr ρn = 1, and tr ρ̇n = 0, we obtain the equation of motion for
ρn by tracing out all other sites:
ρ̇n = −i[ωn an† an ,ρn ] + κ1 (2an† ρn an − an an† ρn − ρn an an† )
+ κ2 2an2 ρn an†2 − an†2 an2 ρn − ρn an†2 an2
+ V [2an ρn an† − an† an ρn − ρn an† an + A(an† ρn − ρn an† )
− A∗ (an ρn − ρn an )],
1 am .
A=
N m
(25)
(26)
These are self-consistent equations, since each oscillator
depends on the mean field A, which itself depends on the
oscillators. There are N such equations.
Now we write out Eq. (25) in terms of matrix elements.
Unlike in Sec. III, we let κ2 be large but finite, and we include
|2 in addition to |0 and |1. Inclusion of |2 is crucial to have a
phase transition, as we show below. Using the notation ρn,j k ≡
j |ρn |k, the equations of motion for the diagonal elements
are
ρ̇n,00 = −2κ1 ρn,00 + 4κ2 ρn,22
IV. MANY QUANTUM OSCILLATORS
Now we study the all-to-all model defined by Eqs. (14)
and (15). The goal is to see whether a synchronization
transition occurs in the quantum limit, and if so, how the
critical coupling Vc for the transition depends on the frequency
− V (Aρn,01 + A∗ ρn,10 − 2ρn,11 ),
(27)
ρ̇n,11 = 2κ1 ρn,00 − 4κ1 ρn,11 + V [−2ρn,11 + 4ρn,22
√
√
+ A(ρn,01 − 2ρn,12 ) + A∗ (ρn,10 − 2ρn,21 )], (28)
022913-5
TONY E. LEE, CHING-KIT CHAN, AND SHENSHEN WANG
PHYSICAL REVIEW E 89, 022913 (2014)
ρ̇n,22 = 4κ1 ρn,11 − 4κ2 ρn,22
+ V (−4ρn,22 +
√
2Aρn,12 +
√ ∗
2A ρn,21 ).
(29)
The equations of motion for the off-diagonal elements
are
√
ρ̇n,10 = (−3κ1 − iωn )ρn,10 + V [−ρn,10 + 2 2ρn,21
−
ρ̇n,21
√
2A∗ ρn,20 + A(ρn,00 − ρn,11 )],
(30)
√
= (−2κ2 − iωn )ρn,21 + 2 2κ1 ρn,10 − 2κ1 ρn,21
+ V [−3ρn,21 + A∗ ρn,20 +
√
√
+ A( 2ρn,10 − ρn,21 )],
(32)
with similar equations for ρn,01 , ρn,12 , and ρn,02 . The mean
field can be written as
A=
√
1 (ρm,10 + 2ρm,21 ).
N m
(33)
We emphasize that Eqs. (27)–(33) are accurate only to O(1/κ2 )
due to truncating the Hilbert space.
(34)
ρ̄11 =
κ1 (κ2 + V )
,
κ12 + κ1 (3κ2 + V ) + V (κ2 + V )
(35)
ρ̄22 =
κ12
,
κ12 + κ1 (3κ2 + V ) + V (κ2 + V )
(36)
C. Stability analysis
Now we do a linear stability analysis of the unsynchronized
state. If the unsynchronized state becomes unstable for certain
parameter values, that signals a continuous phase transition to
the synchronized state since then |A| > 0. We consider small
perturbations δρn around the fixed point: ρn (t) = ρ̄ + δρn (t).
We expand Eqs. (27)–(33) around the fixed point, keeping
only terms to linear order in δρn . It turns out that Eqs. (30)
and (31) decouple from the other equations after linearization,
so we only have to consider them since A depends only on
ρn,10 and ρn,21 :
√
˙ n,10 = (−3κ1 − V − iωn )δρn,10 + 2 2V δρn,21
δρ
˙ n,21
δρ
B. Unsynchronized state
The model in Eq. (14) has a U (1) symmetry: an → an eiβ .
When the coupling V is larger than a critical value Vc , this
symmetry is broken. |A| acts as an order parameter for the
phase transition: |A| = 0 in the unsynchronized state, and
|A| > 0 in the synchronized state.
We now find the unsynchronized state, denoted by ρ̄n ,
which is a fixed point of the mean-field equations but with
no off-diagonal elements, since this implies the lack of
phase coherence among the oscillators. It turns out that ρ̄n
is independent of ωn , so we write ρ̄ ≡ ρ̄n . This state is easily
found by solving for the diagonal elements in Eqs. (27)–(29):
2κ1 κ2 + V (κ2 + V )
,
κ12 + κ1 (3κ2 + V ) + V (κ2 + V )
and all off-diagonal elements are zero so that A = 0. Note that
ρ̄ exhibits amplitude death since a † a → 0 as V → ∞.
2A(ρn,11 − ρn,22 )], (31)
ρ̇n,20 = (−κ1 − 2κ2 − 2iωn )ρn,20 + V [−2ρn,20
ρ̄00 =
+ V A(ρ̄00 − ρ̄11 ),
(37)
√
= 2 2κ1 δρn,10 + (−2κ1 − 2κ2 − 3V − iωn )δρn,21
√
+ 2V A(ρ̄11 − ρ̄22 ),
(38)
A=
√
1 (δρm,10 + 2δρm,21 ).
N m
(39)
We want to find whether the unsynchronized state is stable
or not, i.e., whether the perturbations δρn grow or decay. To
do this, we write δρn (t) = eλt bn , where λ is an eigenvalue and
bn is a 2N -dimensional eigenvector. But we only care about
when an eigenvalue λ = 0, since that corresponds to when the
unsynchronized state just becomes unstable2 ; i.e., V = Vc . So
we solve for bn,10 and bn,21 when λ = 0:
AVc [(2κ1 + 2κ2 + 3Vc + iωn )ρ̄00 − (2κ1 + 2κ2 − Vc + iωn )ρ̄11 − 4Vc ρ̄22 ]
,
8κ1 Vc − (3κ1 + Vc + iωn )(2κ1 + 2κ2 + 3Vc + iωn )
√
2AVc [2κ1 ρ̄00 + (κ1 + Vc + iωn )ρ̄11 − (3κ1 + Vc + iωn )ρ̄22 ]
=−
,
8κ1 Vc − (3κ1 + Vc + iωn )(2κ1 + 2κ2 + 3Vc + iωn )
bn,10 = −
(40)
bn,21
(41)
A=
√
1 (bm,10 + 2bm,21 ).
N m
(42)
2
There is the possibility that the imaginary part of λ is nonzero when the real part is zero. However, numerically we have found that this is
never the case.
022913-6
ENTANGLEMENT TONGUE AND QUANTUM . . .
PHYSICAL REVIEW E 89, 022913 (2014)
Now, we require self-consistency by plugging Eqs. (40) and (41) into Eq. (42). This provides an implicit expression for Vc
in terms of the other parameters. At this point, it is more convenient to move to a continuum description in order to calculate Vc
analytically.
D. Continuum description
So far, we have let oscillator n have frequency ωn and density matrix ρn . Since we are interested in the limit N → ∞, we
can just label the density matrices by frequency, ρn → ρ(ω), where ρ(ω) should be viewed as the average density matrix for all
oscillators with frequency ω. Then bn → b(ω) and
AVc [(2κ1 + 2κ2 + 3Vc + iω)ρ̄00 − (2κ1 + 2κ2 − Vc + iω)ρ̄11 − 4Vc ρ̄22 ]
,
8κ1 Vc − (3κ1 + Vc + iω)(2κ1 + 2κ2 + 3Vc + iω)
√
2AVc [2κ1 ρ̄00 + (κ1 + Vc + iω)ρ̄11 − (3κ1 + Vc + iω)ρ̄22 ]
,
b21 (ω) = −
8κ1 Vc − (3κ1 + Vc + iω)(2κ1 + 2κ2 + 3Vc + iω)
∞
√
A=
[b10 (ω) + 2b21 (ω)]g(ω)dω,
b10 (ω) = −
(43)
(44)
(45)
−∞
where we use the notation bj k (ω) ≡ j |b(ω)|k. Equation (45) shows how the frequency disorder g(ω) comes in.
For self-consistency, we plug Eqs. (43) and (44) into
Eq. (45). We assume that g(ω) is even: g(−ω) = g(ω). Then
since the imaginary parts of b10 (ω) and b21 (ω) are odd in ω,
only their real parts contribute to the integral. Thus, Eq. (45)
becomes
1=
∞
−∞
z 1 ω 2 + z2
g(ω)dω,
ω 4 + z3 ω 2 + z4
(46)
where we have defined the constants
z1 = Vc [(−κ1 + Vc )ρ̄00 + (5κ1 + 4κ2 + Vc )ρ̄11
− (4κ1 + 4κ2 + 2Vc )ρ̄22 ],
z2 = Vc 6κ1 (κ1 + κ2 ) + Vc (3κ1 + 2κ2 ) + 3Vc2
(47)
Delta function (identical oscillators):
g(ω) = δ(ω),
Vc =
g(ω) = 1/(2)
(48)
Vc =
z3 = 13κ12 + 8κ1 κ2 + 4κ22 + 34κ1 Vc + 12κ2 Vc + 10Vc2 ,
2
z4 = 6κ1 (κ1 + κ2 ) + Vc (3κ1 + 2κ2 ) + 3Vc2 .
(49)
(50)
Equation (46) is one of the main results of this paper. It provides
an implicit expression for Vc in terms of the other parameters
and g(ω).
(52)
10κ1 κ2 + 2 +
for ω ∈ [−,],
100κ12 κ22 + 28κ1 κ2 2 + 4
6κ1
(53)
,
(54)
where is the half width of g(ω). Clearly, Vc increases as increases, reflecting the fact that the greater the disorder is,
the harder it is to synchronize the oscillators. Figure 3(a) plots
Eq. (54).
Lorentzian distribution:
g(ω) =
Vc =
E. Results for different disorder distributions
After plugging in a function for g(ω) into Eq. (46) and
doing the integral via contour integration, we can then solve
explicitly for Vc . In general, the result of the integral is very
complicated, but since we only care about the limit of large
κ2 and thus large Vc , we can simplify it by expanding in 1/κ2
and 1/Vc . Here we state the results for different types of g(ω),
valid for large κ2 :
10κ2
.
3
We see that Vc → ∞ as κ2 → ∞ due to too much quantum
noise, so there is no phase transition when the oscillators are
confined to |0 and |1. But there is a transition when κ2 is large
but finite so that the oscillators also occupy |2. (Recall from
Sec. III that for two oscillators, phase correlations do survive
with only |0 and |1.)
It is interesting to note that the phase transition in this
case is due only to dissipative dynamics, instead of a
combination of dissipative and coherent dynamics as in other
models [20,43,49,50].
Uniform distribution:
×[(6κ1 + 2κ2 + 3Vc )ρ̄00 + (−2κ2 + 3Vc )ρ̄11
− (6κ1 + 6Vc )ρ̄22 ],
(51)
1
,
2
π ω + 2
⎧ 2κ (5κ +)
2
1
⎨ 3(κ
1 −)
⎩
∞
(55)
< κ1
,
(56)
κ1
where is the half width at half maximum of g(ω).
Figure 4(a) plots Eq. (56). Interestingly, synchronization never
occurs when κ1 due to the long tails of the Lorentzian
distribution. If the tails are cutoff beyond some value, Vc no
longer diverges, as also shown in Fig. 4(a).
022913-7
TONY E. LEE, CHING-KIT CHAN, AND SHENSHEN WANG
338
336
334
332
330
0.06
6
(a)
Vc (units of κ1)
Vc (units of κ1)
340
PHYSICAL REVIEW E 89, 022913 (2014)
0
1
2
3
(a)
|A|
4
0.04
0.02
2
0
4
Γ (units of κ1)
(b)
0
0
1
2
3
300
350
4
Γ (units of κ1)
400
450
500
550
400
450
500
550
400
450
500
550
V (units of κ1)
0.06
(b)
FIG. 3. Phase diagram for uniform frequency disorder, showing
critical coupling Vc vs. disorder width . (a) Quantum model with
κ2 = 100κ1 using Eq. (54). (b) Classical model using Eq. (58).
|A|
0.04
0.02
0
There is an important difference between the uniform and
Lorentzian cases. In the limit of large κ2 , Vc () for the uniform
case is independent of , because quantum noise is more
important than the disorder. But Vc () for the Lorentzian case
is always dependent on , because the long tails cause the
disorder to be as important as quantum noise.
To check these predictions, we have simulated the meanfield equations Eqs. (27)–(32) using fourth-order Runge-Kutta
integration for N = 3000 with a step size of dt = 5 × 10−4 /κ1
for a time of t = 103 /κ1 . Figure 5 shows that the simulations
agree well with the analytical predictions.
F. Comparison with classical results
We want to compare the dependence of Vc on g(ω) in the
quantum limit with that in the classical limit. Since the classical
model [Eq. (7)] has been studied previously [17], below we
quote the known results for different g(ω), using the definitions
in Eqs. (51), (53), and (55).
Delta function (identical oscillators):
Vc = 0.
2
(a)
Vc (units of κ1)
Vc (units of κ1)
2000
1500
1000
500
0
0
0.25
0.5
0.75
Γ (units of κ1)
1
(b)
1.5
1
0.5
0
0
0.25
0.5
0.75
Γ (units of κ1)
1
FIG. 4. Phase diagram for Lorentzian frequency disorder, showing critical coupling Vc vs. disorder width . (a) Quantum model
with κ2 = 100κ1 . The solid line is without a cutoff [Eq. (56)]. The
dashed line is with a cutoff at |ω| = 100. (b) Classical model using
Eq. (59).
350
V (units of κ1)
0.06
(c)
|A|
0.04
0.02
0
300
350
V (units of κ1)
FIG. 5. (Color online) Synchronization phase transition found
by simulating mean-field Eqs. (27)–(32) with N = 3000 and κ2 =
100κ1 . Order parameter |A| is plotted as a function of coupling
strength V for different frequency distributions. (a) Delta-function
distribution (identical oscillators). (b) Uniform distribution with
= 20κ1 . (c) Lorentzian distribution with = 0.7κ1 and cutoff
at |ω| = 100. Arrows point to the critical coupling predicted by
Eq. (46).
Uniform distribution:
2
2(Vc − κ1 )
= π + tan−1
Vc
= tan−1
Vc
Vc − κ1
(57)
Due to the lack of disorder and noise, there is always
synchronization as long as V > 0.
300
π
κ1
2
π
κ1 .
2
<
(58)
Vc is found by solving these implicit expressions. It is plotted
in Fig. 3(b).
Lorentzian distribution:
⎧
√ 2
2
⎪
⎨ κ1 +3− κ1 −2κ1 +5
< κ1
2
Vc =
.
(59)
⎪
⎩
∞
κ1
This is plotted in Fig. 4(b).
Clearly, the classical results are different from the quantum
results. However, there are also some notable similarities. In
both quantum and classical limits, there cannot be synchronization with Lorentzian disorder when κ1 . Also, for
uniform disorder with large , both quantum and classical
limits have Vc = 2 /3κ1 .
In fact, the quantum results are similar to what one would
expect by adding a lot of white noise to the classical model
[Eq. (7)]. The curve Vc () has nonanalytic points in the
classical limit, but they are smoothed out by quantum noise.
022913-8
ENTANGLEMENT TONGUE AND QUANTUM . . .
PHYSICAL REVIEW E 89, 022913 (2014)
Also, Vc in the quantum limit is a lot larger than in the classical
limit, because the system needs to overcome substantial
quantum noise.
V. CONCLUSION
We have studied the synchronization of dissipatively
coupled van der Pol oscillators in the quantum limit. Synchronization survives all the way down to the quantum limit
(|1 for two oscillators, and |2 for many oscillators), and the
synchronization behavior is qualitatively consistent with noisy
classical oscillators. However, the quantum model exhibits
entanglement, which is absent in the classical model.
Given that synchronization and entanglement can exist in
the quantum limit, there are numerous directions for future
work. One possibility is to go beyond the all-to-all coupling
and consider low-dimensional lattices with short-range coupling. Classical oscillators with short-range coupling exhibit
universal scaling: the correlation length has a power-law
dependence on the disorder width [51]. There can also be phase
transitions in low dimensions [52,53]. One should investigate
how quantum mechanics affects these classical results. A
convenient technique for studying disordered oscillators in
low dimensions is real-space renormalization group, which
was used in the classical regime [51,54,55].
Another promising direction is to characterize the entanglement when there are more than two oscillators. For example,
how does entanglement depend on the frequency disorder?
Does entanglement reach a maximum or exhibit a diverging
length scale at a phase transition [56,57]? Is there multipartite
entanglement [28]?
Aside from synchronization, classical oscillators also
exhibit a variety of collective behavior: glassiness [58],
chimeras [59], phase compactons [60], and topological defects [53]. One should see what happens to these behaviors in
the quantum limit. Furthermore, it would be interesting to put
quantum oscillators on a complex network and see how the
network topology affects the dynamics [61].
ACKNOWLEDGMENTS
This work was supported by the NSF through a grant to
ITAMP. C.K.C. is supported by the Croucher Foundation.
momentum. If the state of the oscillator is given by a density
matrix ρ, the corresponding Wigner function is [47]
1 ∞
dy x − y|ρ|x + y e2ipy , (A1)
W (x,p) =
π −∞
=
m|ρ|nWmn (x,p),
(A2)
mn
where m and n denote Fock states, and we define
1 ∞
Wmn (x,p) =
dy ψm (x − y)ψn (x + y)e2ipy . (A3)
π −∞
ψn (x) is the Fock state |n in the position basis:
14
1
x2
ψn (x) =
Hn (x).
exp −
π 4n (n!)2
2
(A4)
Hn (x) is the Hermite polynomial of degree n.
Then a two-mode Wigner distribution W (x1 ,p1 ,x2 ,p2 )
for two oscillators can be thought of as a quasiprobability
distribution in the four-dimensional phase space. It is defined
by
∞
1
W (x1 ,p1 ,x2 ,p2 ) = 2
dy1 dy2 e2i(p1 y1 +p2 y2 )
π −∞
× x1 − y1 ,x2 − y2 |ρ|x1 + y1 ,x2 + y2 ,
=
(A5)
m1 m2 |ρ|n1 n2 Wm1 n1 (x1 ,p1 )Wm2 n2 (x2 ,p2 ). (A6)
m1 n1 m2 n2
Now we change variables a few times. First, √
we write the
Wigner function
in
terms
of
α
=
(x
+
ip
)/
2 and αn∗ =
n
n
n
√
(xn − ipn )/ 2 instead of xn and pn , where αn corresponds
to a coherent state. Then we move to polar coordinates: αn =
rn eiθn . Then since we only care about the relative phase θ =
θ2 − θ1 , we integrate out r1 , r2 , and θ1 + θ2 . After doing all
this, we find that if the density matrix is
⎞
⎛
0
0
0
f1
f2
g + ih 0 ⎟
⎜0
(A7)
ρ=⎝
0 g − ih
f3
0⎠
0
0
0
f4
in the basis {|00,|01,|10,|11}, the corresponding two-mode
Wigner function is
Here, we review what a two-mode Wigner distribution is
and provide details on how to derive Eq. (23). First, recall
that the one-mode Wigner function W (x,p) for an oscillator
can be thought of as a quasiprobability distribution in the
two-dimensional phase space, where x is position and p is
g cos θ + h sin θ
1
+
.
(A8)
2π
2
This is how we derived Eq. (23).
√
Now suppose the system is in |S = (|01 + |10)/ 2. The
corresponding√W (θ ) is peaked at θ = 0. In contrast, |A =
(|01 − |10)/ 2 has W (θ ) peaked at θ = π . This is why
|S and |A correspond to in-phase and antiphase locking,
respectively.
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TONY E. LEE, CHING-KIT CHAN, AND SHENSHEN WANG
PHYSICAL REVIEW E 89, 022913 (2014)
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